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Boolean Algebra

Boolean Algebra. Logical Statements. A proposition that may or may not be true: Today is Monday Today is Sunday It is raining. Compound Statements. More complicated expressions can be built from simpler ones: Today is Monday AND it is raining. Today is Sunday OR it is NOT raining

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Boolean Algebra

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  1. Boolean Algebra

  2. Logical Statements • A proposition that may or may not be true: • Today is Monday • Today is Sunday • It is raining

  3. Compound Statements • More complicated expressions can be built from simpler ones: • Today is Monday AND it is raining. • Today is Sunday OR it is NOT raining • Today is Monday OR today is NOT Monday • (This is a tautology) • Today is Monday AND today is NOT Monday • (This is a contradiction) • The expression as a whole is either true or false.

  4. Things can get a little tricky… • Are these two statements equivalent? • It is not nighttime and it is Monday OR it is raining and it is Monday. • It is not nighttime or it is raining and Monday AND it is Monday.

  5. Boolean Algebra • Boolean Algebra allows us to formalize this sort of reasoning. • Boolean variables may take one of only two possible values: TRUE or FALSE. • (or, equivalently, 1 or 0) • Algebraic operators: + - * / • Logical operators - AND, OR, NOT, XOR

  6. Logical Operators • A AND B is True when both A and B are true. • A OR B is always True unless both A and B are false. • NOT A changes the value from True to False or False to True. • XOR = either a or b but not both

  7. Writing AND, OR, NOT • A AND B = A ^ B = AB • A OR B = A v B = A+B • NOT A = ~A = A’ • TRUE = T = 1 • FALSE = F = 0

  8. Exercise • AB + AB’ • A AND B OR A AND NOT B • (A + B)’(B) • NOT (A OR B) AND B

  9. Boolean Algebra • The = in Boolean Algebra means equivalent • Two statements are equivalent if they have the same truth table. (More in a second) • For example, • True = True, • A = A,

  10. Truth Tables • Provide an exhaustive approach to describing when some statement is true (or false)

  11. Truth Table

  12. Truth Table

  13. Truth Table

  14. Truth Table

  15. Example • Write the truth table for A(A’ + B) + AB’ • Fill in the following columns: • A, B, A’, B’, A’ + B, AB’, A (A’ + B), whole expression.

  16. A (A’ + B) + AB’

  17. Exercise • Write the truth table for (A + A’) B

  18. Solution to (A + A’) B

  19. A + True = True A + False = A A + A = A A + B = B + A (commutative) A AND True = A A AND False = False A AND A = A AB = BA (commutative) Boolean Algebra - Identities

  20. Associative and Distributive Identities • A(BC) = (AB)C • A + (B + C) = (A + B) + C • A (B + C) = (AB)+(AC) • A + (BC) = (A + B) (A + C) • Exercise: using truth tables prove - • A(A + B) = A

  21. Solution: A AND (A OR B) = A

  22. Using Identities • A + (BC) = (A + B)(A + C) • A(B + C) = (AB) +(AC) • A(A + B) = A • A + A = A • Exercise - using identities prove: • A + (AB) = A • A +(AB) = (A +A)(A + B) • = A (A + B) = A

  23. Identities with NOT • (A’)’ = A • A + A’ = True • AA’ = False

  24. DeMorgan’s Laws • (A + B)’ = A’B’ • (AB)’ = A’ + B’ • Exercise - Simplify the following with identities • (A’B)’

  25. Solving a Truth Table

  26. Exercise: Solving a Truth Table

  27. Exercise: Solving a Truth Table

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